The Dynamics of Carbon Sequestration and Alternative Carbon
Accounting, with an Application to the Upper Mississippi River Basin
Hongli Feng
Working Paper 05-WP 386
March 2005
Center for Agricultural and Rural Development
Iowa State University
Ames, Iowa 50011-1070
www.card.iastate.edu
Hongli Feng is an associate scientist at the Center for Agricultural and Rural Development, Iowa State
University.
This paper is available online on the CARD Web site: www.card.iastate.edu. Permission is granted to
reproduce this information with appropriate attribution to the author.
For questions or comments about the contents of this paper, please contact Hongli Feng, 560D Heady
Hall, Iowa State University, Ames, IA 50011-1070; Ph: 515-294-6307; Fax: 515-294-6336; E-mail:
hfeng@iastate.edu.
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The Dynamics of Carbon Sequestration and Alternative Carbon
Accounting, with an Application to the Upper Mississippi River Basin
Abstract: Carbon sequestration is a temporal process in which carbon is continuously being stored/released over a period of time. Di erent methods of carbon
accounting can be used to account for this temporal nature, including annual average
carbon, annualized carbon, and ton-year carbon. In this paper, starting by exposing
the underlying connections among these methods, we examine how the comparisons
of sequestration projects are a ected by these methods and the major factors a ecting them. We explore the empirical implications for carbon sequestration policies by
applying these accounting methods to the Upper Mississippi River Basin, a large and
important agriculture area in the United States. We nd that the di erences are signi cant in terms of the location of land that might be chosen and the distribution of
carbon sequestration over the area, although the total amount of carbon sequestered
does not di er considerably across programs that use di erent accounting methods
or di erent values of the major factors.
Keywords: annual average carbon, annualized carbon, carbon sequestration,
ton-year carbon.
JEL Classi cation: Q20, Q25
1
Introduction
Carbon sequestration through land use changes and forestry has been the focus
of considerable attention in the climate change literature because of its potential as
a cost-e ective mitigation strategy. With the Kyoto Protocol becoming a binding
treaty, countries may have further incentives to incorporate it into their greenhouse
gas management plans. Carbon sequestration is a temporal process that removes
carbon from the atmosphere either evenly or unevenly over time: the amount of
carbon removal is larger in some periods than in others. Negative sequestration,
that is, carbon release into the atmosphere is also possible over some time intervals
even though a project has overall positive sequestration. In order to properly assess
di erent sequestration projects, it is critical that this temporal attribute be properly
accounted for. In this paper, we examine some of the important issues related to
1
carbon accounting and its policy implications when sequestration becomes part of
the climate change mitigation portfolio.
In reporting the amount of carbon that has been sequestered in a project, several accounting methods and their variations have been used or proposed in the
literature, including the annual average carbon, the annualized carbon, and ton-year
carbon. Simply speaking, the annual average carbon, the most widely used accounting method, is the sum of total carbon sequestered over a xed period of time divided
by the length of the period. To re ect our preference for bene ts that have occurred
earlier, the annualized carbon accounting method discounts carbon sequestered later.
Although new relative to annual average carbon, annualized carbon (or its variation,
the present discounted value of carbon) has been employed by many studies, including
Adams et al. (1999), Plantinga et al. (1999), and Stavins (1999). A third accounting
method, the ton-year carbon, takes into account the duration of carbon kept outside of the atmosphere. Several studies (e.g., Watson et al., 2000; Moura-Costa and
Wilson, 2000) have analyzed this method with an emphasis on how it facilitates the
comparison among projects that sequester (or release) carbon for di erent lengths of
time.
Di erent projects may show up as the favorable choices when di erent accounting
methods are used. Even under the same accounting method, the ranking of projects
may di er, as the value of some factors varies. The rst factor is the project duration.
There might be some natural choices for the value of this factor, for example, the
saturation point, which is the length of time needed for a carbon pool to reach
equilibrium. Given that there may be di erent carbon pools in a single project (let
alone in multiple projects), the use of saturation point may result in (a) di erent
durations in di erent projects and (b) a somewhat subjective decision on which, if
2
any, carbon pool's saturation point to use. In fact, di erent durations of projects have
been employed in the literature to suit the underlying nature of the analyses. For
example, Stavins (1999) used a period of 90 years to allow at least one rotation of each
project species; Parks and Hardie (1995) limited their study to the life of a temperate
forest; and Adams et al. (1999) chose a 50-year period to investigate the costs of
sequestration through both a orestation and improvement in forest management.
The e ect of the choice of project durations is largely determined by the path of
a sequestration project (i.e., distribution of carbon sequestration over time), which
is the second factor we are going to explore. Obviously, an accounting method that
gives more weight to early sequestration will favor a project that sequesters carbon
in relatively early periods. Although seldom discussed in the literature on the cost of
sequestration, the e ects of di erent mitigation paths have been extensively debated
in the more general climatic change literature (see Wigley et al., 1996 and Ramakrishna, 1997). Some have argued that delaying abatement may be costly because there
is socioeconomic inertia in the energy system and the process of climate change is
di cult to reverse. If earlier carbon sequestration is valued more, then we may prefer
one sequestration project over another even if both projects can sequester the same
amount of carbon (undiscounted sum) at the same amount of cost over the same
period of time. To take into account the timing of carbon uptake, discounting can
be used.
The discount rate is the third factor that we are going to discuss. Instead of
sensitivity-type analysis, we examine how the discount rate interacts with sequestration paths and project durations to a ect the results of sequestration policies. The
advantage of discounting is that it can re ect preferences for early carbon reduction
and allow us to focus on some summary measures (e.g., annualized carbon) without
3
being too concerned about the paths of sequestration. However, discounting also
brings its own complications because, as we illustrate, a di erent discount rate may
favor a di erent sequestration activity and, even at the same discount rate, di erent
projects may become the favorable choice as project durations vary.
Some studies have started investigating the issue of accounting for time in climate
mitigation through carbon sequestration. The di erences of alternative accounting
methods and the factors a ecting them are discussed in the special report on Land
Use, Land-Use Change and Forestry by the Intergovernmental Panel on Climate
Change (IPCC) (Watson et al., 2000, Chapters 2 and 5). Fearnside et al. (2000)
indicated that temporal issues will be key factors in determining whether e orts to
mitigate global warming will include carbon sinks and which sequestration activities
should be given priority. They identi ed ton-year carbon accounting as a mechanism
to compare sequestration of di erent durations. They also showed that discounting
can strongly in uence economic decisions. Tipper and De Jong (1998) carefully
described the accounting methods that we analyze in this paper and discussed their
advantages and disadvantages. They also applied their discussions to a project in
Mexico.
Building on previous research, we contribute to the literature by (i) investigating
how the comparisons of the accounting methods are a ected by three factors (project
durations, discount rate, and carbon sequestration paths) and how the factors interact
with each other, and (ii) applying the analysis to a major agricultural area in the
United States and demonstrating the di erent outcomes of policies based on di erent
accounting methods or di erent values of the factors. We start by describing the
accounting methods and their underlying connections in the next section. Then in
sections 3-4 we analyze the e ects of the three factors. The empirical application is
4
provided in section 5. The last section gives concluding comments.
2
The alternative accounting methods
In order to understand the underlying connections between the accounting meth-
ods, we begin with the following framework. Consider a program with a funding of
M; which selectively enrolls elds from a total of N agricultural or forest elds to
sequester carbon from time 0 to time T: We call T the project duration. Each piece
of land is enrolled for some carbon sequestration practices during this period of time.
The size of eld n is denoted as An ; where n is the index of a eld. Carbon sequestered
at time t by a unit of land on eld n is denoted as xn (t)
0: Denote the cost of
enrolling a unit of land from eld n as pn (t); which is the pro t forgone and/or establishment expenditures due to the adoption of carbon sequestration activities. Given
that our focus is on carbon accounting, for simplicity we assume that the bene t of
carbon sequestration for mitigating climate change is constant over time, denoted as
b for any t for each unit of carbon sequestered.
Suppose the policymaker's problem is to choose an for each eld to maximize the
present discounted value (PDV) of the bene t from sequestration over the project
duration, i.e.,
max
an
s:t:
N Z
X
n=1 0
N
X
T
Z
T
b an xn (t) e
rs
dsdt
(1a)
t
an
Z
T
e
rt
pn (t)dt = M;
(1b)
An ;
(1c)
0
n=1
0
an
where r is the discount rate for the policymaker. In the objective function, the inner
integration represents the overall bene t from the carbon sequestrated at time t in
eld n. The bene t starts accruing from the time of sequestration (t) and lasts
5
until the end of the project duration (T ). The outer integration essentially sums
up the bene t of carbon sequestered at each point of time over [0; T ]. The budget
constraint, (1b), indicates that the total payment over [0; T ] for all enrolled elds
is equal to the available funding.1 It is important to note here that all enrollment
occurs at time 0 and an does not change with time. Because of the reversibility of
carbon sequestration, it is important for a policy to enroll a farmer for a relatively
long period of time. Also for the same reason of non-permanence, it is important to
take into account the duration of bene ts from sequestration.
Using
as the multiplier for the budget constraint, the solutions to (1) can be
written as follows:
8 9
<An =
0
an =
: ;
a
where
8 9
Z T
<>=
if bXn <
e rt pn (t)dt;
: ;
0
=
Z T
Z T
Xn
xn (t)
e rs dsdt; and
RT
P
a
i
i:
a
=A
0 e
f i ig
rt p
i (t)dt
(3)
t
0
a = M
(2)
.R
T
0
e
rt p
n (t)dt
: Intuitively,
is the shadow
cost of funding. The left-hand side of the conditions in (2) represents the total PDV
of bene t from eld n; while the right-hand side represents the PDV of costs from
eld n for carbon sequestration over the project duration. Whether to enroll eld n
depends on whether the PDV of bene t is greater than the PDV of cost. If bene t is
less than cost, then the eld will not be enrolled. On the contrary, a whole eld will
be enrolled if its bene t is greater than the corresponding cost. When bene t and
cost are equal for a eld, the area enrolled from the eld (a ) is determined by the
price of enrolling the parcel and the funding left after payment for all other parcels
with whole- eld enrollment.
The term Xn can be considered as a measure of total carbon accumulated on
1
Another form of resource constraint is the total available land. However, our discussions would
mostly stay the same with the use of this constraint.
6
one unit of
eld n. In empirical analysis, several important simpli cations of Xn
are often used, including the sum of carbon, the PDV of carbon, and the ton-year
carbon. By dividing a constant (some variation of the length of time), these measures
are equivalent to annual average carbon, annualized carbon, and the average ton-year
carbon, respectively. We will next de ne these terms, show their connections with
Xn ; and highlight the assumptions under each of these measures.
De nition 1 The sum of carbon sequestration in a unit of land on eld n (Xn ) is
the simple summation of carbon sequestered over [0; T ]; and annual average carbon
(xn ) is the sum of carbon sequestered divided by the corresponding period of time;
that is,
Z
Xn
T
xn (t)dt; and xn
0
Xn
:
T
(4)
Comparing Xn and Xn ; we know that the latter is derived by setting r = 0 and
assuming that carbon sequestered at time t only has e ect at time t (as opposed to
over the period from t to T ): Thus, the sum of carbon does not take into account
the fact that, relative to carbon sequestered later, carbon sequestered earlier provides
earlier bene t that is usually valued more and provides bene t for a longer duration.
De nition 2 The present discounted value of carbon sequestration in a unit of land
^ n ) is the sum of carbon sequestered over [0; T ] weighted by a discounting
on eld n (X
term, e
rt ;
and the annualized carbon (^
xn ) is a constant equal for all time points
^ n ; that is,
such that the PDV of this constant is equal to X
^n
X
Z
T
e
rt
xn (t)dt; and x
^n
0
RT
0
^n
X
e
rt dt
:
(5)
^ n , we have X
^ n = Xn : Also, if we assume that carbon seSetting r = 0 in X
^ n . It may seem unusual
questered at time t only has e ect at time t; then Xn = X
7
to use the PDV of a physical good, carbon, because PDVs are in general calculated
for monetary values. In fact, in the model setup (1), only monetary values are discounted. Carbon discounting results from the discounting of the bene t provided by
future sequestration.
~ n ) is
De nition 3 The ton-year carbon sequestration in a unit of land on eld n (X
the sum of carbon sequestered over [0; T ] weighted by the length of the period lasting
from the time carbon is sequestered until the end of the project duration,2 and average
ton-year carbon (~
xn ) is a constant equal for all time points such that the ton-year
~ n ; that is,
carbon of this constant is equal to X
~n
X
Z
0
T
Z
t
T
~n
X
xn (t)dsdt; and x
~n = R T R T
:
0 t 1dsdt
(6)
~ n and Xn is the discounting
It is easy to see that the only di erence between X
~ n and X
^ n (or
factor: speci cally, in the former, r is set to 0: One di erence between X
Xn ) is that the former takes into account the duration of bene t while the latter does
not. It is interesting to note that some researchers have proposed a conversion factor
between 1 ton-year carbon of sequestration and 1 ton of carbon emission reduction,
which is estimated to be about 0.0182 (Moura-Costa and Wilson, 2000; and Tipper
and De Jong, 1998).
From their de nitions, it is clear that the accounting methods can be very different mainly due to some factors: r; T; and xn (t). The sequestration path, xn (t), is
given for a xed eld with speci ed land use practices except for variations caused
by natural uncertainties. As we discussed in the introduction, the choice of T is
not as obvious as it rst appears. When projects have di erent sequestration paths,
the choice of r and T may a ect the ranking of projects no matter which carbon
2
Although in our analysis ton-year carbon does not incorporate discounting, one might modify
the approach to include discounting.
8
accounting method is used. As a result, di erent projects may be included in carbon
sequestration policies, which in turn may a ect the overall outcome of policies.
3
The e ects of discounting under di erent carbon paths
If carbon sequestration is linear (i.e., constant over time), say, xn (t) = x; then it
is easy to show that all three accounting methods will be equivalent in the sense that
the annual average carbon, the annualized carbon, and the average ton-year carbon
will have exactly the same value. In such cases, the choice of projects is not a ected
by the choice of accounting methods. However, carbon sequestration is usually nonlinear. For example, carbon sequestration by a orestation is largely determined by
the accumulation of biomass, which is generally known to be slow at the beginning,
faster in the midterm, and then slow again near maturity. The duration of each stage
could range from a few years to more than a hundred years, depending on the timber
species. Two examples are illustrated in Figure 1, which is based on Richards et
al. (1993). A similar process also exists for carbon sequestration by switching from
conventional tillage to conservation tillage (Lal et al., 1998).
When linearity is not satis ed, two elds may have quite di erent annualized
carbon even if they have the same undiscounted annual averages of carbon and the
same r and T are used. In other words, x
^n
if xn
x
^m may be large in absolute value even
xm is zero. The disparity between x
^n
x
^m and xn
xm is a ected by the
curvature of xn (t) and xm (t), since x
^n and x
^m discount later sequestration while xn
and xm do not. If we view carbon sequestration xn (t) as the weight attached to the
corresponding time t; then xn (t); upon appropriate normalization, can be viewed as
the probability density function of t. This view of xn (t) enables utilization of wellknown results in the literature on nance and risk. Before invoking any result, we
9
present the de nitions of two concepts.
De nition 4 Let F i (y) and F j (y) be two cumulative distribution functions (cdf 's) of
a random variable y 2 y; y : (i) F i (y) rst-order stochastically dominates (FOSD)
F j (y) if
Z
y
Z
i
(y)dF (y)
y
y
(y)dF j (y)
(7)
y
for any non-decreasing function (y): (ii) F i (y) second-order stochastically dominates
(SOSD) F j (y) if (7) holds for any non-decreasing concave function (y):
We can show that F i (y) FOSD F j (y) if F i (y)
F j (y); for any y 2 y; y :
Also, given the same mean, if Y has larger variability under F j (y) than under F i (y)
then F i (y) SOSD F j (y): Loosely speaking, FOSD compares the means (levels) of
two distributions while SOSD, in addition, compares their spread over the domain
of a random variable. Suppose the two cdf's are the distributions of random net
returns associated with two investment projects. If F i (y) FOSD F j (y); that is, higher
returns are more likely to occur under F i (y) than under F j (y); then the former will
be preferred over the latter by any investor who values higher returns more. If F i (y)
SOSD F j (y); then the former will be preferred over the latter by risk-averse investors
because net returns from the former tend to be less variable and/or higher in all
states of nature.
In our context, we can construct a cdf as follows. De ne
Rt
xi (s)ds
i
; i = 1; 2; :::; N:
F (t) = R 0T
0 xi (s)ds
(8)
Although F i (t) satis es all the conditions required for a cdf, di erent probability
densities are only arti cially attached to di erent values of t since t is not a random
variable. Based on F i (t), we next provide a proposition on the comparison of carbon
paths based on annual average carbon and annualized carbon.
10
Proposition 1 Given xn = xm ; if F m (t) rst-order stochastically dominates F n (t)
then x
^n
x
^m for any r > 0; if F m (t) second-order stochastically dominates F n (t),
then x
^n
x
^m for r > 0:3
A proof is given in the appendix. Intuitively, the probability density arti cially
given to each value of t is determined by the rate of carbon sequestration. The
rst-order stochastic dominance by F m (t) over F n (t) means that proportionally less
carbon is accumulated earlier under path xm (t) than under path xn (t): Given that
earlier sequestration is valued more (for r > 0) in calculating the PDV of a stream
of carbon sequestration, the PDV is greater from xn (t) than from xm (t); that is,
^n
X
^ m ; or equivalently, x
X
^n
x
^m :
For the paths given in Figure 1, the two pines have about the same annual
average carbon, 2.15 tons/year/acre, for the period from year 0 to year 77. Then
based on the curvature of the two paths, it is easy to see that F m (t)
stochastically dominates F n (t) for the same period. So x
^n
rst-order
x
^m for any r > 0 by
Proposition 1. In fact, for the same period and at a 2 percent discount rate, the
annualized carbon sequestration for loblolly and ponderosa pines are x
^n = 2:62 and
x
^m = 1:84 tons/year/acre, respectively. The di erence between x
^n
x
^m and xn
xm
is about 0:78 tons/year/acre, which accounts for 36 percent of the annual average
carbon sequestration.
When there is no FOSD relationship between F m (t) and F n (t); the rst half
of Proposition 1 does not apply. However, if we know that F m (t) SOSD F n (t);
then we can invoke the second half of the proposition. Graphically, the second-order
stochastic dominance of F m (t) over F n (t) implies that carbon uptake spreads out
3
From their de nitions, it is clear that FOSD implies SOSD. That is, if F m (t) FOSD F n (t); then
F (t) SOSD F n (t): Thus, the rst part of the proposition is captured in the second part of the
proposition. Both are presented here to facilitate discussion.
m
11
more evenly over time and/or occurs earlier along path xn (t) than along path xm (t)
(see Figure 2). Because of discounting, the value of carbon sequestration decreases at
an exponential rate, e
rt ,
and so the annualized carbon is higher for a carbon path
with relatively more early carbon sequestration. Thus, we have x
^n
x
^m :
It is important to note that Proposition 1 only applies to some carbon paths
since FOSD and SOSD do not completely characterize the relationship between two
cdf's. It may happen that neither cdf dominates the other in terms of FOSD or
SOSD. In these situations, the comparisons of accounting methods and carbon paths
will be more complicated, as illustrated by the following section.
4
The e ects of T, r, and carbon sequestration path
In this section, we will explore by illustration how T , r, and sequestration paths
a ect the accounting methods, focusing on how the comparisons of the accounting
methods might be a ected by the factors and how the e ects of one factor might
be in uenced by another factor. Tables (1a) and (1b) show annualized carbon sequestration (^
xn ) in an acre of a orestation for two species of pines with di erent r
and T. The rst row of the tables (with r = 0) indicates the e ect of di erent T on
^ n ; we use its normalized version x
the annual average carbon. Instead of X
^n to make
meaningful comparisons because it is hard to make sense of the comparison between
carbon sequestered, say, over 20 years and over 50 years, unless we take into account
the length of time. The two pines are used as an illustration because of the sharp
contrast in their sequestration paths as indicated by Figure 1.
From Tables (1a) and (1b), we can see that for loblolly pine, as T increases, the
annualized carbon decreases for all four discount rates shown,4 while for ponderosa
4
At r = 0:15 for T > 60; the change in x
^n is negligible. This is because r is so high that little is
added to the numerator and denominator of (5) as T increases. This is also the case for ponderosa
pine.
12
pine, it rst increases and then decreases. Similarly, as r increases, for ponderosa
pine the annualized carbon decreases for all four project durations, while for loblolly
pine the relationship varies with T: The lack of a speci c pattern in the change
of annualized carbon in response to the change of T (or r) can be explained by
the de nition of x
^n in (5). For any given r; as T increases x
^n will also increase
if xn (T )
x
^n (a term in d^
xn =dT ) is positive. Intuitively, this implies that if the
carbon sequestration rate is higher at T or after T than the annualized sequestration
rate over [0; T ]; then as T increases, x
^n becomes larger. This is the case when T
increases from 30 years to 60 or 90 years for ponderosa pine. On the other hand, if
the sequestration rate decreases over time, then the di erence of the two terms will
be negative and x
^n will decrease as T increases. This is the case for loblolly pine
when T is greater than 30 years or for ponderosa pine when T increases from 90 to
160 years.
Similarly, we can explain the phenomena that, as r increases, x
^n may increase for
one sequestration path but not the other and the trend even varies for the same given
path and project duration as in the case of loblolly pine. As r increases, both the
RT
RT
numerator, 0 e rt xn (t)dt; and the denominator, 0 e rt dt; in (5) will decrease. If
the former decreases more slowly (quickly) than the latter, then x
^n will increase (deRT
RT
RT
R
rt x (t)dt= T e rt x (t)dt is positive
crease). That is, if 0 te rt dt= 0 e rt dt
n
n
0 te
0
(negative), then x
^n will increase (decrease). The rst ratio is essentially the average
of t weighted by e
e
rt x (t).
n
rt
and the second ratio is essentially the average of t weighted by
If the rate of sequestration tends to be higher closer to time T , then the
second ratio gives relatively more weight to larger t; which implies the di erence of
the two ratios will be negative and x
^n will decrease with r:
Intuitively, as r increases, the value of later sequestration will be valued even less,
13
which implies lower annualized carbon if sequestration tends to occur later. This is
the case for all listed project durations for ponderosa pine, which has an increasing
rate of sequestration for a long period of time (nearly 80 years). For loblolly pine,
although sequestration starts to decline around year 20, the rate of sequestration is
still relatively high for the period between year 20 and year 30. Thus, over the period
from year 0 to year 30, more sequestration occurs relatively later, and so x
^n decreases
as r increases. However, for other T values in Table (1a), more sequestration occurs
relatively earlier and so the value of x
^n increases for low discount rates. At the
highest discount rate in the table (r = :15); the trend is reversed. The reason is that,
at a very large r; the value of e
rt
decreases rapidly and so only sequestration that
occurs really early matters. This means that the increasing trend in the early years of
loblolly pine dominates the decreasing trend later. As a result, x
^n actually decreases.
By comparing the values in Tables (1a) and (1b), it is obvious that loblolly pine
has higher annualized carbon than the ponderosa pine for all T and r except for
a very low discount rate and long project duration, that is, (r = 0; T = 90); and
(r = 0; T = 160): While not shown, it is not necessarily true that, for a given T value,
high discount rates always favor one project while low discount rates favor another.
More speci cally, in order to compare the annualized carbon sequestration along two
di erent paths xn (t) and xm (t) for the same duration; we can assess the sign of
D(r)
Z
T
e
rt
[xn (t)
xm (t)] dt:
0
From the de nition of x
^n in (5), we know x
^n
we treat xn (t)
x
^m
0 if and only if D(r)
0: If
xm (t) as the net cash ow of a project, then D(r) is the net present
discounted value of this project. The discount rate r such that D(r) = 0 is called the
internal rate of return (ROR) of a project. The well-known disadvantage of ROR is
that it is not unique and that D(r) is not monotone in r: Saak and Hennessy (2001)
14
and Oehmke (2000) identi ed conditions in which D(r) is monotone in r:
Even when the dividing discount rate such that D(r) = 0 is unique, it can
change as project duration varies. In the example of the loblolly pine and ponderosa
pine, at T = 160; the dividing discount rate is about 0:0153; that is, the PDV of
carbon sequestration by loblolly pine is larger if r > 0:0153: At T = 90; the dividing
discount rate is about 0:0074; which is smaller than that at T = 160: This is because
carbon sequestration by loblolly pine almost tapers o
to zero for T > 90; while
sequestration by ponderosa pine is still signi cant (see Figure 1). Thus, for a project
duration longer than 90 years, a higher discount rate is needed in order for the PDV
of carbon sequestration by loblolly pine to remain larger.
Tables 2a and 2b show the average ton-year carbon (~
xn ) with di erent T for
the same two pines. As T increases, x
~n shows about the same trend as x
^n with
r = 0 (which is also xn , shown in the rst row in Tables (1a) and (1b)). The only
di erence is in ponderosa pine going from T = 90 to T = 160: x
~n is increasing while
xn is decreasing. This can be explained by the following di erence in the accounting
methods. As T increases, how x
~n changes depends on the balance of two e ects: the
longer duration of early sequestered carbon and the carbon sequestered after T: The
rst e ect is not re ected in xn : If too little carbon is sequestered after T , there will
be a decreasing pressure on x
~n : If the increasing e ect from the longer duration of
early sequestration cannot outweigh this decreasing e ect; then x
~n decreases, as in
the case of loblolly pine for all project durations and ponderosa pine when T increases
from 90 to 160: However, if there is enough sequestration after T; combined with the
increasing e ect of the increased duration of early sequestration, then this means that
x
~n increases, as in the case of ponderosa pine for T = 30; 60; and 90:
15
5
Implications for sequestration policies
In this section, we examine the policy implications of the accounting methods
and the factors a ecting them for an important region of the United States, the Upper Mississippi River Basin (UMRB). The UMRB covers 492,000 square kilometers
mostly in ve states (Illinois, Iowa, Missouri, Minnesota, and Wisconsin) of the central United States. This area is comprised of 131 U.S. Geological Service (USGS)
eight-digit watersheds (shown on Figures 3-5) and is dominated by agriculture: cropland and pasture together account for about 67 percent of the total area. Our primary
data source is the latest available National Resource Inventory (NRI) (USDA-NRCS,
1997) which provides information on the natural resource characteristics of the land,
cropping history, and farming practices across the region. To estimate the environmental bene ts, we use the Environmental Policy Integrated Climate (EPIC) model
version 3060.5 EPIC simulations were run for each NRI point in the region (over
40,000 points in total) for the speci ed project duration.
We consider a green payment type policy; that is, we assume that policymakers
pay farmers to adopt conservation practices on their elds to sequester carbon. The
conservation practice we consider here is no-till, which has been shown to have carbon
sequestration potential (Lal, et al. 1998). We assume that the goal of the policymakers is to maximize the bene t from carbon sequestration as speci ed in equation (1a)
with simpli cations as implied by De nitions 1-3 to account for di erent accounting
methods. With a constraint of a given total acreage of land enrolled (speci cally, 20
percent of the UMRB cropped area excluding pasture), it is optimal for policymakers
to pay for elds with the highest carbon sequestration potential to adopt no-till.
The alternative policy scenarios we consider di er only by the accounting mecha5
EPIC has been tested under a variety of conditions. Additional information concerning EPIC
and details concerning model assumptions and data can be found in Feng et al. 2004.
16
nism used to measure sequestration potential. For example, if sum of carbon is used,
then elds with the highest sum of carbon over the project duration will be selected
rst, regardless of these elds' sequestration potential in terms of ton-year carbon, or
the PDV of carbon. Similarly, if ton-year carbon is used, then elds will be ranked
by their sequestration potential in terms of ton-year carbon over the project duration
and the elds ranked on top will be enrolled into the program, regardless of these
elds' sum of carbon or PDV of carbon over the same period.
In the following, we present the results of some pair-wise comparisons of the
alternative accounting mechanisms.6 While these comparisons use a project duration
of 20 years, we also illustrate the e ects of project duration by comparing policies
which rank elds based on 50-year and 20-year sum of carbon, respectively. For most
elds, rapid carbon sequestration occurs within the rst 20 years after switching to no
till and a new soil carbon equilibrium will be reached by the 50th year. However, there
is a large degree of heterogeneity due to the variations in crop rotation, soil, and other
natural conditions. The di erences between the policy scenarios can be illustrated in
two ways: the location of elds enrolled and the amount of carbon sequestered. To
represent the di erence in the location of land enrolled, we rst compute, for each
8-digit USGS watershed in the UMRB, the area of all elds that are only enrolled
under one policy (y1 ) and the area of all elds that are enrolled under both policies
(y2 ), and then calculate the ratio, 100
y1 =y2 . To illustrate the di erence between
the amount of carbon sequestered under the policy using one accounting method (y3 )
and the carbon sequestered under the policy using another accounting method (y4 ),
we use the percentage di erence, that is, 100 (y4
y3 )=y3 .
Figures 3-5 display these di erences: the deeper the color the larger the di er6
To avoid redundancy, only a few pair-wise comparisons are chosen, even though there are potentially many such comparisons.
17
ence. A few observations can be made based on the maps. First, in terms of either
location of land enrolled or carbon sequestered, the di erence between the accounting
methods using 20-year and 50-year annual average carbon appears to be the largest
and the di erence between ton-year accounting and PDV accounting methods the
smallest (see Figure 3 versus Figure 4). This indicates that many elds that have
relatively high annual average carbon over 20 years do not have high annual average carbon over 50 years. On the other hand, a eld that has high ton-year carbon
also tends to have high PDV of carbon over the same period, probably because both
accounting methods give more weight to early sequestration (although for di erent
reasons and in di erent ways).
Second, the di erences are larger when expressed in the form of the location of
land enrolled than in the form of carbon sequestered, as illustrated by the contrast
between maps on the left and the corresponding maps on the right in Figures 3-5.
This is mainly because (1) the di erence between the elds in terms of carbon sequestration potential is small, and (2) the percentage di erences are based on the same
accounting method in order to make the comparisons meaningful, even though the
policies are based on di erent accounting methods. Third, for all of the comparisons,
the di erences tend to concentrate on Iowa and southern Minnesota where most land
is enrolled. Fourth, in term of the magnitude of di erence, 73 percent (40 percent) of
the watersheds have a di erence larger than 25 percent (50 percent) in terms of the
location of elds in the comparison between the 20-year and 50-year annual average
carbon, the comparison showing the largest di erences (see Figure 3A). Interestingly,
in terms of the total sum of carbon sequestered over the whole UMRB, the di erence between the policies is almost negligible (and so not presented here). It is up
to policymakers to decide in the actual design of policies (a) the importance of the
18
aggregate sequestration potential, and (b) the importance of heterogeneity in terms
of geographical location.
6
Conclusions
Because of the dynamics of carbon sequestration, the accounting for time in the
estimate of carbon storage is critical in assessing sequestration projects and in comparing carbon sinks and other climate change mitigation options. The time dimension
of carbon sequestration is accounted for in di erent ways in the di erent accounting methods discussed in this paper. In analyzing sequestration options, the choice
of accounting mechanisms tends to be study speci c. The annual average carbon
method and a default project duration of 20 years are currently used in IPCC's good
practice guidance on inventorying and reporting greenhouse gas emissions and removals in \cropland remaining cropland" and \land converted to cropland" (Penman
et al., 2003). Given that di erent projects might be favored under di erent accounting mechanisms, regions/countries may advocate an accounting system that is most
suitable for their sequestration projects. For example, regions with relatively early
sequestration may advocate annualized carbon because of its preferential treatment
for early sequestration.
The quantity of carbon sequestered is not the only consequence of the use of
alternative accounting systems. In fact, in our empirical analysis, no signi cant
di erence is found in terms of total amount of carbon sequestered among policies using
di erent accounting mechanisms. Instead, we nd that quite di erent geographical
areas will bene t under the policies. Our results may be speci c to our study region
and the sequestration activities considered. However, this points out a possibility
that governments can choose an accounting mechanism to meet other policy goals
such as income support for a certain group of people. Of course, how much freedom
19
a national government has in choosing accounting methods depends on the extent of
international coordination.
Appendix: Proof for Proposition 1
Note that (t) =
e
rt
is an increasing function for any r > 0. Thus, if F m (t)
FOSD F n (t); then by (7) we have
Z
T
( e
rt
)dF m (t)
0
Z
T
( e
rt
)dF n (t);
0
8 r > 0:
(A-1)
Plugging in (8) and rearranging, we obtain
Z
0
T
e
RT
0
rt x (t)
m
xm (s)ds
Given xn = xm ; we know
RT
dt
Z
T
0
e
RT
0
rt x (t)
n
xn (s)ds
dt;
8 r > 0:
RT
xn (s)ds. Thus we can drop them from
RT
both sides of the above inequality. Then, dividing both sides by 0 e rt dt, we have
0
xm (s)ds =
x
^m
Similarly, since (t) =
e
rt
0
x
^n ;
8 r > 0:
is non-decreasing and concave for any r > 0; (A-1) still
holds if F m (t) SOSD F n (t): The second half of the proof follows in a way similar to
the proof for the rst half of Proposition 1.
20
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21
Table 1. Annualized carbon sequestration (^
xn ) (tons/acre/year)7
(a) Loblolly pine
r=0
r=.02
r=.05
r=.15
T=30
3.562
3.485
3.333
2.704
T=60
2.621
2.885
3.083
2.700
(b) Ponderosa pine
T=90
1.868
2.479
2.979
2.700
T=160
1.065
2.163
2.948
2.700
r=0
r=.02
r=.05
r=.15
T=30
1.369
1.261
1.101
0.694
T=60
1.992
1.731
1.365
0.713
T=90
2.224
1.889
1.418
0.713
T=160
1.940
1.878
1.415
0.713
Table 2. Average ton-year carbon (~
xn ) (tons/acre/year)8
(a) Loblolly pine
T=30
3.338
T=60
3.130
T=90
2.607
(b) Ponderosa pine
T=160
1.754
T=30
1.009
7
T=60
1.568
T=90
1.887
T=160
2.058
For easy reference: 1 acre=0.40 hectare, 1 ton=0.91 tonne, and 1 ton/acre=2.24 tonne/hectare.
Had we used discounting in our de nition of ton-year carbon, we could obtain tables like Tables
(1a) and (1b) with similar results. Results are available from the author upon request.
8
22
Figure 1. The path of loblolly and pondorosa pines
Figure 2. Comparison of carbon paths
23
Figure 3. Project duration of 20 years vs project duration of 50 years9
A. Di erence in location of land enrolled
B. Di erence in carbon sequestered
Figure 4. Ton-year carbon vs PDV of carbon accounting methods
A. Di erence in location of land enrolled
9
B. Di erence in carbon sequestered
In all maps, `No Data' indicates that no area is chosen in the sub-watershed.
24
Figure 5. Ton-year carbon vs sum of carbon accounting methods
A. Di erence in location of land enrolled
25
B. Di erence in carbon sequestered